Inverse Limit of Sheaves

Let $X = \text{Spec}\;ℂ[x]$, and $\mathcal{G}⊂\mathcal{O}_X$ be the subsheaf of regular (rational) functions with no pole at $0$ or $∞$, and at most simple poles elsewhere. Thus if $U = ℂ \backslash \lbrace a_1,\dotsc,a_m \rbrace$ or $ℂ \backslash \lbrace a_1,\dotsc,a_m,0 \rbrace$, with $a_1,\dotsc,a_m$ distinct and nonzero, then $\mathcal{G}(U) = \lbrace\dfrac{f}{∏_{i=1}^m(x-a_i)} \vert \deg(f)≤m \rbrace$. Then let $\mathcal{G}_n = \bigoplus_{d=1}^n\mathcal{G}$, with $\mathcal{G}_{n+1}↠\mathcal{G}_n$ by projection onto the first $n$ components. Let $\mathcal{F}_n = \bigoplus_{d=1}^n\mathcal{F}^d$, where $\mathcal{F}^d⊂\mathcal{G}$ is the subsheaf of functions with a zero of order at least $d$ at $0$. Then the sheaf (not presheaf!) $\mathcal{G} / \mathcal{F}^d$ is isomorphic to the constant sheaf $ℂ[x]/(x^d)$. Therefore for nonempty $U$, $(\varprojlim\;\mathcal{G}_n/\mathcal{F}_n)(U) = ∏_{d=1}^∞ ℂ[x]/(x^d)$, whereas $\varprojlim\;\mathcal{G}_n(U) = ∏_{d=1}^∞ \mathcal{G}(U)$. Passing to stalks, we see $\varprojlim\;\mathcal{G}_n → \varprojlim\;\mathcal{G}_n/\mathcal{F}_n$ is not surjective, since $ℂ(x) → ℂ[[x]]$ is not.

Remark: Obviously this example is not as simple or enlightening as yours. However, one can argue that the example needs to be at least this complex, I believe. Without going into details, let me at least say that surjectivity of a sheaf map is a local property, so it takes more than a global topological obstruction to block it.

A natural example of this is the inverse system $\left \{ \mathbb Z/ \ell^n \mathbb Z \right \}_{n\ge 1}$ of $\ell-$adic sheaves in the étale topology. In this case $\lim^1 \left \{ \mathbb Z/ \ell^n \mathbb Z \right \}_{n\ge 1} \ne 0 $.

However, the same example does not work in the Zarisky topology (by flasqueness of the sheaves involved) or in the pro-étale topology (by the existence of weakly contractible objects), i.e., it holds for two different reasons: sheaves are not interesting enough (Zarisky) and the topology is very good (pro-\'etale).